We have seen that \(\delta \) -SMI and \(\eta \) -SMI can give very similar posteriors, identical in the simple normal example in Section REF , and in general depending on the chosen \(\delta \) -SIM smoothing kernel \(K_\delta \) . We presented SMI as examples of Gibbs posteriors with loss functions which are only prequentially additive but give valid and order-coherent belief updates. The \(\eta \) -SMI family of posterior distributions are based on power posteriors. This is a natural choice, but not the only one available. Any order-coherent family of distributions interpolating Cut and Bayes is potentially of interest. One good feature of \(\delta \) -SMI posteriors is that \(\delta \)  has the same dimension as the data \(Y\) ,
so the measure of misspecification has a simple interpretation. It is large or small compared to the variation in the sampled \(Y\) -values. Also, the \(\delta \) -SMI posterior is a kind of ABC-posterior in which we condition on the data in some neigbourhood of the observed data. However, in \(\delta \) -SMI this neighborhood is a product space of neighborhoods for each observation \(Y_i,\ i=1,...,n\) , there is no “summary statistic" and we recover Bayesian inference as \(\delta \rightarrow 0\) .
This kind of connection between ABC-like methods and misspecification has been noted elsewhere [1]}.
